Biharmonic operator on a manifold

Question

On curved surfaces the Laplacian is replaced by the Laplace-Beltrami operator: $$ \nabla^2 = \frac{1}{\sqrt{g}}\partial_i\sqrt{g}g^{ij}\partial_j$$ How to calculate higher powers of the Laplace Beltrami operator, particularly in relation to the Biharmonic operator on a curved space, where the biharmonic operator is $\nabla^4f=\nabla^2(\nabla^2f)$ ?